Multiplication operators: domain, spectral measure and spectrum

Multiplication by $m$ is well defined on null classes: two representatives that agree off a null set have products agreeing there, and the squared-integrability condition is unchanged. It is linear on its domain, since $\mathrm{\mid }m(u+v){\mathrm{\mid }}^2\le 2\mathrm{\mid }mu{\mathrm{\mid }}^2+2\mathrm{\mid }mv{\mathrm{\mid }}^2$. For $u\in H$, $u_n=u{\mathbf{1}}_{\{\mathrm{\mid }m\mathrm{\mid }\le n\}}$ belongs to the domain and tends to $u$ in $L^2$ by domination by $\mathrm{\mid }u{\mathrm{\mid }}^2$, so the domain is dense. All multiplications and norms below use the complex Hilbert structure in [A1].

A bounded measurable multiplier $a$ gives a bounded operator with $\mathrm{\parallel }au{\mathrm{\parallel }}_2\le (\sup \mathrm{\mid }a\mathrm{\mid })\mathrm{\parallel }u{\mathrm{\parallel }}_2$. Define $E(B)u={\mathbf{1}}_{m^{-1}(B)}u$ for Borel $B\subseteq \mathbb{R}$. It is idempotent and self-adjoint by the integral pairing in [A1]. Preimages show normalization and multiplicativity. For disjoint Borel $B_j$, the difference between $E({\bigcup }_j{B}_j)u$ and its first $n$ summands has squared norm the integral of $\mathrm{\mid }u{\mathrm{\mid }}^2$ over the remaining preimages, tending to zero by dominated convergence. Thus $E$ is a PVM.

If $H=\{0\}$, sigma-finiteness forces $\mu =0$: otherwise some finite-measure set in a countable finite-measure cover would have positive measure, and its indicator would be a nonzero $L^2$ vector. All operators then have full zero-space domain and zero action, and the spectrum and essential range are empty; [A4] and [A5] give the direct zero-space conventions. All claims follow in this case. Henceforth assume $H\mathrm{\ne }\{0\}$ before using the bounded calculus or spectral uniqueness in [A4]–[A5].

Its scalar measure is $E_u(B)={\int }_{m^{-1}(B)}\mathrm{\mid }u{\mathrm{\mid }}^{2}d\mu$, a finite Borel measure of mass $\mathrm{\parallel }u{\mathrm{\parallel }}_2^2$. The real line is Hausdorff (disjoint small intervals separate distinct points), locally compact by compact closed bounded intervals, and second-countable by the rational base in [A3]. Therefore the regularity theorem in [A3] applies to every $E_u$: $E$ is regular. Moreover, for any nonnegative Borel $q$, $\int qd{E}_u=\int (q\circ m)\mathrm{\mid }u{\mathrm{\mid }}^{2}d\mu$. This holds first for indicators by the displayed scalar measure, then finite nonnegative simple sums, then all nonnegative Borel functions by increasing simple approximation and monotone convergence.

For a complex Borel simple function $s$ on $\mathbb{R}$, complete its disjoint representation with the zero-coefficient complement. By [A4], its integral against $E$ is multiplication by $s\circ m$. Given bounded Borel $h$, partition a square containing its complex range into finitely many Borel cells of diameter tending to zero, taking a fixed corner as each coefficient; these give complex simple $s_n$ with $\sup \mathrm{\mid }{s}_n-h\mathrm{\mid }\to 0$. The multiplier norm bound from step 1.2 and the operator-norm approximation in [A4] imply ${\mathrm{\Phi }}_E(h)=M_{h\circ m}$.

For arbitrary Borel $g:\mathbb{R}\to \mathbb{C}$, step 2.1 with $q=\mathrm{\mid }g{\mathrm{\mid }}^2$ identifies the domain of $g(E)$ with $\{u\in H:\int \mathrm{\mid }g\circ m{\mathrm{\mid }}^2\mathrm{\mid }u{\mathrm{\mid }}^{2}d\mu <\mathrm{\infty }\}$. Its bounded truncations act by $g(m){\mathbf{1}}_{\{\mathrm{\mid }g(m)\mathrm{\mid }\le n\}}u$ by step 2.2, and these tend in $L^2$ to $(g\circ m)u$ by dominated convergence. In particular $\int \lambda dE$ equals $M_m$ with exactly the stated domain. Regularity proved in step 2.1 licenses the converse spectral theorem in [A5], so $M_m$ is self-adjoint and $E$ is its spectral PVM, unique among regular representing PVMs. Thus the calculation for general $g$ is indeed its functional calculus.

For any measurable set $A$, the indicator multiplier $M_{{\mathbf{1}}_A}$ vanishes if $\mu (A)=0$. Conversely, let $(X_n)$ be a countable cover by finite-measure sets. If $\mu (A)>0$, some $A\cap X_n$ has positive finite measure, since a countable union of null sets is null. Then $u={\mathbf{1}}_{A\cap X_n}$ is a nonzero $L^2$ vector fixed by that multiplier. Therefore $E(B)=0$ exactly when $\mu (m^{-1}(B))=0$. Apply the spectral essential-range formula in [A5] to the identity function: this gives precisely the stated real essential range of $m$. Nonreal points are outside that range since the PVM is on the real line. This completes the domain, self-adjointness, spectral measure, calculus and spectrum claims.